Thermal Product Formula in Holography

• Thermal product formula is a representation of finite-temperature correlators defined by quasi-normal mode (QNM) poles and normalization, exhibiting meromorphy and the absence of zeros.
• It employs Hadamard factorization to express the correlator as an infinite product, linking ultraviolet OPE constraints with the infrared hydrodynamic expansion.
• Extensions to mixed shear channels and reflecting cavity settings highlight its universality in capturing QNM effects, bouncing geodesics, and black-hole thermodynamic insights.

The thermal product formula is a product representation of finite-temperature correlators in which holographic thermal two-sided two-point functions are determined by their quasi-normal mode (QNM) spectrum, up to normalization. In its canonical form, the frequency-space correlator is meromorphic, has no zeros, and therefore admits Hadamard factorization as an infinite product over QNM frequencies. Subsequent work extended this structure to mixed shear channels in charged AdS black branes and to reflecting-cavity settings for asymptotically flat and de Sitter black holes, where it links QNM asymptotics to “bouncing geodesics” and singularities of thermal correlators. In a distinct black-hole thermodynamics literature, “product formula” instead refers to products of horizon entropies or other horizon quantities; there, thermal and logarithmic corrections generally spoil classical mass-independence and quantization (Dodelson et al., 2023, Bhattacharya et al., 24 Apr 2025, Grozdanov et al., 9 Jun 2026, Pradhan, 2016).

1. Product representation of thermal two-point functions

In large $N$ strongly coupled quantum systems with holographic duals, the central statement is that the two-sided thermal two-point function $G_{12}(\omega)$ is a meromorphic function of $\omega$, its singularities are the QNMs, it has no zeros in the complex $\omega$-plane, and its asymptotics are such that $1/G_{12}(\omega)$ is entire of order one. These analytic properties imply, by the Hadamard factorization theorem, that the correlator is fixed by its poles alone together with a normalization constant (Dodelson et al., 2023).

The resulting factorization is

$$G_{12}(\omega)=\frac{G_{12}(0)}{\displaystyle\prod_{n=1}^{\infty}\left(1-\frac{\omega^2}{\omega_n^2}\right)\left(1-\frac{\omega^2}{(\omega_n^*)^2}\right)} .$$

Here $\omega_n$ denotes a QNM frequency in one quadrant of the complex plane, while the factors involving $\pm\omega_n$ and $\pm\omega_n^*$ implement the symmetry of the spectrum. For purely imaginary modes, the product is written as $\prod_n (1+\omega^2/\tilde{\omega}_n^2)^{-1}$. The same analysis gives the relation

$G_{12}(\omega)$0

so the pole structure of the two-sided correlator and of the retarded function is essentially the same.

This factorization is not merely a formal rewriting. It states that the entire thermal two-point function is encoded in the QNMs. The paper further emphasizes that the product formula gives a natural dispersive representation with a positive discontinuity at the location of QNMs. A plausible implication is that the formula reorganizes real-time thermal response in terms of the damped collective excitations of the black-hole background.

2. Constraints from OPE, black-hole singularities, and hydrodynamics

The product formula is constrained by three inputs: the operator product expansion (OPE), the singularity inside the black hole, and the hydrodynamic expansion. At large real frequency, the OPE gives the asymptotic behavior

$G_{12}(\omega)$1

which requires an infinite tower of QNMs extending to large $G_{12}(\omega)$2. For a single asymptotic line of QNMs,

$G_{12}(\omega)$3

the OPE constrains the spacing through

$G_{12}(\omega)$4

Thus the ultraviolet structure of the field theory fixes asymptotic features of the QNM distribution (Dodelson et al., 2023).

The black-hole interior supplies an independent constraint. In AdS$G_{12}(\omega)$5 black holes, the singularity controls the large imaginary-frequency behavior,

$G_{12}(\omega)$6

which leads to dispersive sum rules of the form

$G_{12}(\omega)$7

for odd $G_{12}(\omega)$8. Deforming the contour yields

$G_{12}(\omega)$9

with $\omega$0. These relations enforce exponential decay of the residues.

At low frequency, the hydrodynamic expansion is expressible in terms of moments of the QNM distribution: $\omega$1 The moments are positive, and the paper notes that this positivity constrains the allowed region for hydrodynamic coefficients. This suggests that the thermal product formula functions simultaneously as an analyticity statement, a spectral representation, and a bridge between ultraviolet OPE data and infrared transport data.

3. Mixed channels and the sum-of-products generalization

The original factorization was formulated for scalar fluctuations. In the shear channel of R-charged black branes in five-dimensional AdS, however, the shear mode is coupled with the charge diffusion mode at non-zero momentum. When these modes are suitably decoupled into master variables, an exact formula again exists for the two-point functions of the boundary current and energy-momentum tensor in terms of QNMs, but the structure is no longer a single product. Instead, the physical correlator becomes a weighted sum of product-form contributions (Bhattacharya et al., 24 Apr 2025).

For each decoupled master variable $\omega$2, the Wightman function takes the product form

$\omega$3

The boundary correlator is then

$\omega$4

This “sum-of-products” structure is the main extension beyond the scalar-channel formula. For $\omega$5, corresponding to the case without a bulk Chern-Simons term, there are two master variables. For $\omega$6, where a boundary global R-symmetry anomaly is present and the shear channel acquires additional couplings, four master variables are required.

The same work reports two dynamical consequences of the QNM structure. First, for sufficiently large Chern-Simons coefficient $\omega$7, certain QNMs move into the upper-half complex plane, signaling an instability. Second, at large momentum and large $\omega$8, some QNMs approach the real axis, producing high-momentum long-lived modes. In the extremal limit $\omega$9, the near-horizon AdS$\omega$0 analysis yields scaling exponents

$\omega$1

with $\omega$2 determined by effective AdS$\omega$3 masses; violation of the AdS$\omega$4 Breitenlohner-Freedman bound matches the instability region found at nonzero temperature. This establishes that, in mixed channels, the thermal product formula persists in modified form and remains tightly linked to the QNM spectrum.

4. Cavity thermal product formula and bouncing geodesics

In asymptotically flat and de Sitter black holes there is no canonical timelike boundary analogous to AdS. A reflecting cavity supplies such a boundary and makes possible a cavity version of the thermal product formula. The central geometric ingredient is the existence of “bouncing geodesics”: null trajectories that originate outside a black hole, approach its curvature singularity, bounce off infinitesimally close to the singularity, and return. These geodesics generate “bouncing singularities” in the retarded Green’s function, and the cavity thermal product formula establishes a universal relation between the locations of those singularities and the cavity QNM spectrum (Grozdanov et al., 9 Jun 2026).

The asymptotic relation is

$\omega$5

where $\omega$6 is the bouncing time and $\omega$7 is the $\omega$8-th high-overtone cavity QNM frequency. The real part of $\omega$9 is the principal-value coordinate time from wall to singularity and back, while the imaginary part counts the number of horizons crossed en route, measured in units of $1/G_{12}(\omega)$0. The corresponding product formula for the two-sided boundary correlator is

$1/G_{12}(\omega)$1

A key point is that this relation holds for Dirichlet, Neumann, or Robin wall conditions. The paper further gives explicit bouncing times in special cases. For asymptotically flat $1/G_{12}(\omega)$2 Schwarzschild with wall at radius $1/G_{12}(\omega)$3,

$1/G_{12}(\omega)$4

For certain Schwarzschild–de Sitter configurations, closed-form expressions involving the black-hole and cosmological horizon radii $1/G_{12}(\omega)$5 and $1/G_{12}(\omega)$6 are also obtained. Numerical computations of scalar, electromagnetic, and gravitational cavity QNMs show fast convergence of the QNM spacing to $1/G_{12}(\omega)$7, confirming the formula.

This cavity construction extends to $1/G_{12}(\omega)$8 and $1/G_{12}(\omega)$9 a structure previously known in AdS. The reflecting wall is introduced by hand rather than arising as a natural boundary, but with that modification the analytic structure and the link between QNMs and null geodesics generalize across all signs of the cosmological constant.

5. Generality, evidence, and limitations

The 2023 work proposes a broader “thermal product hypothesis”: the two-sided thermal two-point correlator in a chaotic large $$G_{12}(\omega)=\frac{G_{12}(0)}{\displaystyle\prod_{n=1}^{\infty}\left(1-\frac{\omega^2}{\omega_n^2}\right)\left(1-\frac{\omega^2}{(\omega_n^*)^2}\right)} .$$0 system is a meromorphic function with no zeros in the $$G_{12}(\omega)=\frac{G_{12}(0)}{\displaystyle\prod_{n=1}^{\infty}\left(1-\frac{\omega^2}{\omega_n^2}\right)\left(1-\frac{\omega^2}{(\omega_n^*)^2}\right)} .$$1-plane and therefore is given by a product over its poles. Evidence is presented not only in holographic black-hole backgrounds but also in several non-gravitational large $$G_{12}(\omega)=\frac{G_{12}(0)}{\displaystyle\prod_{n=1}^{\infty}\left(1-\frac{\omega^2}{\omega_n^2}\right)\left(1-\frac{\omega^2}{(\omega_n^*)^2}\right)} .$$2 chaotic models, while explicit failures are identified in free or integrable settings (Dodelson et al., 2023, Grozdanov et al., 9 Jun 2026).

The positive examples include the SYK model and the $$G_{12}(\omega)=\frac{G_{12}(0)}{\displaystyle\prod_{n=1}^{\infty}\left(1-\frac{\omega^2}{\omega_n^2}\right)\left(1-\frac{\omega^2}{(\omega_n^*)^2}\right)} .$$3-dimensional SYK chain at $$G_{12}(\omega)=\frac{G_{12}(0)}{\displaystyle\prod_{n=1}^{\infty}\left(1-\frac{\omega^2}{\omega_n^2}\right)\left(1-\frac{\omega^2}{(\omega_n^*)^2}\right)} .$$4. In SYK, the two-sided thermal correlator is known exactly and is meromorphic with poles but no zeros. By contrast, free $$G_{12}(\omega)=\frac{G_{12}(0)}{\displaystyle\prod_{n=1}^{\infty}\left(1-\frac{\omega^2}{\omega_n^2}\right)\left(1-\frac{\omega^2}{(\omega_n^*)^2}\right)} .$$5 SYM at zero coupling and vector large-$$G_{12}(\omega)=\frac{G_{12}(0)}{\displaystyle\prod_{n=1}^{\infty}\left(1-\frac{\omega^2}{\omega_n^2}\right)\left(1-\frac{\omega^2}{(\omega_n^*)^2}\right)} .$$6 models exhibit zeros and branch cuts, so the product formula does not hold. This sharp distinction suggests that meromorphy and absence of zeros are tied to the combination of chaos and large $$G_{12}(\omega)=\frac{G_{12}(0)}{\displaystyle\prod_{n=1}^{\infty}\left(1-\frac{\omega^2}{\omega_n^2}\right)\left(1-\frac{\omega^2}{(\omega_n^*)^2}\right)} .$$7, rather than to large $$G_{12}(\omega)=\frac{G_{12}(0)}{\displaystyle\prod_{n=1}^{\infty}\left(1-\frac{\omega^2}{\omega_n^2}\right)\left(1-\frac{\omega^2}{(\omega_n^*)^2}\right)} .$$8 alone.

The cavity formulation strengthens the claim of universality in another direction. With a reflecting wall, the same basic factorization and high-overtone relation survive in asymptotically flat, de Sitter, and AdS black holes. This suggests that the product formula is controlled less by AdS/CFT in isolation than by a common analytic structure of thermal correlators and QNM spectra when an appropriate boundary observable is available.

A common misconception is that the thermal product formula is simply a restatement of QNM pole expansions. The cited works make a stronger claim: the correlator is fixed by an infinite product over QNMs because it has no zeros and suitable entire-function growth. In mixed channels this becomes a sum of such products, and in cavity settings the same structure encodes information about the singularity through the bouncing time.

6. Distinct usage in black-hole thermodynamics

A different literature uses “product formula” for products of thermodynamic quantities evaluated at the outer and inner horizons of a black hole. In the classical setting, such products can be mass-independent and in some cases quantized. For the Kerr–Newman family,

$$G_{12}(\omega)=\frac{G_{12}(0)}{\displaystyle\prod_{n=1}^{\infty}\left(1-\frac{\omega^2}{\omega_n^2}\right)\left(1-\frac{\omega^2}{(\omega_n^*)^2}\right)} .$$9

and for Kerr,

$\omega_n$0

This mass-independence was often called “universality.” However, once stable thermal fluctuations or statistical quantum fluctuations around thermal equilibrium are included, the corrected entropy of each horizon acquires logarithmic terms, and the entropy product becomes mass-dependent and non-quantized (Pradhan, 2016, Pradhan, 2016).

The basic corrected entropy formula is

$\omega_n$1

and the corrected product is

$\omega_n$2

A CFT derivation yields an equivalent leading-log structure,

$\omega_n$3

Explicit calculations for Reissner–Nordström, Kehagias-Sfetsos, Schwarzschild–AdS, RN–AdS, Kerr, Kerr–Newman, Kerr–Newman–AdS, BTZ, dilaton, Kerr–Sen, Sultana–Dyer, $\omega_n$4, and 5D Gauss–Bonnet black holes all show the same loss of universality.

This usage should not be conflated with the correlator-based thermal product formula. In the entropy-product literature, thermal effects spoil a previously universal horizon product. In the correlator literature, thermal structure produces a product over QNMs. The shared phrase “product formula” therefore covers two conceptually distinct constructions.

Related classical thermodynamic product formulas illustrate the same distinction. For the Kehagias-Sfetsos Hořava-Lifshitz black hole, the horizon radii product, area product, entropy product, and irreducible mass product are universal, while the surface temperature product, Komar energy product, and specific heat product are not universal because they depend on the mass parameter. The same work states that the First law and Smarr-Gibbs-Duhem relations do not hold for this black hole, and that under certain conditions the specific heat signals a second order phase transition (Pradhan, 2015). These results belong to horizon thermodynamics rather than to the QNM-based thermal product formula for correlators.